Indian Statistical Institute8th Mile Mysore Road, Bangalore 560 059, India

url: http://www.isibang.ac.in/∼ statmath

Student’s BrochureB. Math. (Hons.) Programme

Effective from 2015-16 Academic Year

All rules are subject to change and approval by the Dean of Studies, Indian Statistical

Institute, 203, B.T. Road, Kolkata 700108

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Contents

1 General Information 3

1.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Course Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Satisfactory Conduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Examination Guidelines for Students . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Academic Information 5

2.1 Attendance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Class-Teacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Examinations and Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Promotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Repeating a year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Award of Certificates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.8 Fees, Stipend, and Contigency Grant . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 B. Math (Hons.) Curriculum and Detailed Syllabi 12

3.1 Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Sixteen Core courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Statistics courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Computer Science Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 Physics Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Non Credit Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Elective Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Miscellaneous 32

4.1 Library Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Hostel Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Expenses for the Field Training Programmes . . . . . . . . . . . . . . . . . . . . 33

4.4 Change of Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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1 General Information

1.1 Scope

The B. Math (Hons.) degree programme is a premier undergraduate programme in India

that offers comprehensive instruction in mathematics along with courses in Probability, Statis-

tics, Computer Science and Physics. It is so designed that after successfully completing this

programme, a student will be able to pursue higher studies in areas of mathematical sciences

including Mathematics, Statistics, Computer Science, Mathematical Physics and Applied math-

ematics. Students who successfully complete the requirements for the B. Math (Hons.) degree

will automatically be admitted to the M. Math programme of the Indian Statistical Institute.

1.2 Duration

The total duration of the B. Math (Hons.) programme is three years (six semesters). An

academic year, consisting of two semesters with a recess in-between, usually starts in July and

continues till May. The classes are generally held only on the weekdays from 10 am to 5:30 pm.

1.3 Centre

The B. Math (Hons.) programme is offered at the Bangalore centre only.

1.4 Course Structure

The B. Math (Hons.) programme has twenty eight one-semester credit courses and a non-credit

course on Writing of Mathematics, as given in the curriculum below in Section 3.1

1.5 Satisfactory Conduct

The students shall observe all rules (inclusive of hostel and mess rules) of the Institute.

Ragging is banned in the Institute and anyone found indulging in ragging will be given punish-

ment such as expulsion from the Institute, or, suspension from the Institute/classes for a limited

period and fine. The punishment may also take the shape of (i) withholding Stipend/Fellowship

or other benefits, (ii) withholding results, (iii) suspension or expulsion from hostel and the likes.

Local laws governing ragging are also applicable to the students of the Institute. Incidents of

ragging will be reported to the police.

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Students shall not indulge in rowdyism or any other act of indiscipline or unlawful/unethical/

indecent behavior. Attendance requirements in classes detailed in Section 2.1 should be met.

Violations of the above will be treated as breach of discipline and unsatisfactory conduct. They

will attract penalties ranging from : withholding promotion/award of degree, withdrawal of

stipend and/or expulsion from the hostel/Institute.

1.6 Examination Guidelines for Students

1. Students are required to take their seats according to the seating arrangement displayed.

If any student takes a seat not allotted to him/her, she/he may be asked by the invigi-

lator to hand over the answer script and leave the examination hall (i.e., discontinue the

examination).

2. Students are not allowed to carry inside the examination hall any mobile phone with

them, even in switched-off mode. Calculators, books and notes will be allowed inside the

examination hall only if these are so allowed by the examiner(s), or if the question paper

is an open-note/book one. Even in such cases, these articles cannot be shared.

3. No student is allowed to leave the examination hall without permission from the invigila-

tor(s). Further, students cannot leave the examination hall during the first 30 minutes of

any examination. Under no circumstances, two or more students writing the same paper

can go outside at the same time.

4. Students should ensure that the main answer booklet and any extra booklet bear the

signature of the invigilator with date. Any discrepancy should be brought to the notice

of the invigilator immediately. Presence of any unsigned or undated sheet in the answer

script will render it (i.e., the unsigned or undated sheet) to be canceled, and this may

lead to charges of violation of the examination rules.

5. Any student caught cheating or violating examination rules for the first time will get Zero

in that paper. If the first offence is in a backpaper examination the student will get Zero

in the backpaper. (The other conditions for promotion, as mentioned in Section 1.8 in

Students brochure, will continue to hold.)

6. Any student caught cheating or violating examination rules is not eligible for direct ad-

mission to the M. Math programme.

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7. Any student caught cheating or violating examination rules for the second time will be

denied promotion in that year. This means that

(i) a student not already repeating, will have to repeat the corresponding year without

stipend; (ii) a student already repeating, will have to discontinue the programme.

Any student caught cheating or violating examination rules twice or more will be asked

to discontinue the programme and leave the Institute.

2 Academic Information

2.1 Attendance

Every student is expected to attend all the classes. If a student is absent, she/he must apply for

leave to the Associate Dean or Students In-Charge. Failing to do so may result in disciplinary

action. Inadequate attendance record in any semester would lead to reduction of stipend in the

following semester; see Section 2.8.

2.2 Class-Teacher

One of the instructors is designated as the Class Teacher. Students are required to meet

their respective Class Teachers periodically to get their academic performance reviewed, and

to discuss their problems regarding courses.

2.3 Examinations and Scores

There are two formal examinations in each course: mid-semestral (midterm) and semestral

(final).

The composite score in a course is a weighted average of the scores in the mid-semestral

and semestral examinations, home-assignments, quizzes and the practical record book (and/or

project work) in that course. The weights of examinations in a course are announced before

the mid-term examination of the semester by the Students-in-charge or the Class Teacher, in

consultation with the teacher concerned. In the case of courses involving field work, some

weightage is given to the field reports also. The semestral examination has a weight of at least

50%.

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The minimum composite score to pass a credit or non-credit course is 35%.

Back Paper Examination : If the composite score of a student in a course is above 35% but

falls short of 45%, she/he will have an option to take a back-paper examination to improve the

score to a maximum of 45%. This is called an optional back-paper. However, a student with

composite score less than 35% in any course must take a backpaper examination to improve the

score to a maximum of 45%. Such a back-paper is called a compulsory back-paper. At most

one back-paper examination is allowed in a particular course.

The ceiling on the total number of backpaper examinations a student can take is as follows:

4 in the first year, 3 in the second year, 3 in the final year. Note that this ceiling

is for the entire academic year. If a student takes more than the allotted quota of backpaper

examinations in a given academic year, then at the end of that academic year the student should

decide which of the optional back-paper examination scores should be disregarded. In such a

case, the marks of those particular courses will be reverted to their original scores.

Compensatory Examination: The following rule applies to a student who has failed a course (i.e

obtained less than 35% in both the composite score and compulsory back paper examination)

but scores 60% or more in all other courses: If such a student is not in the final year of the

programme, she/he may be provisionally promoted without stipend or contingency grant to

the following year, subject to the requirement that the paper is cleared through the so-called

compensatory examination, which is a regular (semestral) examination in the corresponding

semester of the following year, along with the regular courses for that semester in the current

year. Only the score in the semestral examination need be considered for the purpose of

evaluation. The student is not expected to attend the course, or to take the mid-semestral

examination or to do assignments, projects, etc. even if these are prescribed for the course in

that semester. However, she/he can score at most 35% in such an examination. A student

scoring less than 35% in this examination will have to discontinue the programme, regardless of

the year of study in the programme. Stipend may be restored after she/he successfully clears

the examination, but not with retrospective effect. Also, she/he will not be eligible for any

prizes or awards. All other terms and conditions related to the compensatory examination

remain unchanged. In case the student in question is in the final year of the programme, the

Dean of Studies, in consultation with the Teachers’ Committee, may decide on the mechanism

of conducting a special examination of that particular course along the lines suggested above,

within six months of the end of that academic year.

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Supplementary Examination: In the event of a student failing to take a mid-semestral, semestral,

backpaper or compensatory examination due to medical or family emergencies, the Dean of

Studies or the Students-in-charge, should be informed about this in writing, that is, through an

e-mail or a letter, latest by the date of the examination. Subsequently, she/he should apply to

the aforementioned authorities in writing for permission to take a supplementary examination,

enclosing supporting documents. The supplementary semestral examination is held at the same

time as the back-paper examinations for that semester and a student taking supplementary

semestral examination is not allowed to take a backpaper examination in the course. When

a student takes supplementary semestral (backpaper, compensatory) examination in a course,

the maximum she/he can score in that examination is 60% (45%, 35% respectively). The score

obtained in the supplementary semestral examination will be treated as the semestral exam

score in the calculation of the composite score in that course.

2.4 Promotion

A student passes a semester of the programme only when she/he secures composite score of

35% or above in every course AND his/her conduct has been satisfactory. If a student passes

both the semesters in a given year, the specific requirements for promotion to the following year

are as follows: the average composite score in all the credit courses taken in a year should be

45%, and that the score(s) in non-credit course(s) should be at least 35%.

2.5 Repeating a year

A student fails a year if the student is not eligible for promotion.

Subject to approval of the Teacher’s committee: a student who fails can repeat any

one of the first two years and the final year; a student who secures B. Math degree without

Honours1 and has at most eight composite scores (in credit courses) less than 45% in the first

two years, is allowed to repeat the final year.

The repeat year must be the academic year immediately following the year being repeated. A

repeating student will not get any stipend or contingency grant or prizes during the repeat year.

However, if the student is from such an economically underprivileged background that this step

1See Section 2.6

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will force the student to discontinue, then the student can appeal to the Dean of Studies or the

Students-In-charge, for financial support.

A student repeating a year must be assessed for all courses even if the student has passed them

in the original year, and the student must obtain a minimum of the respective pass marks in

such courses in the repeat year. The final score in a course being repeated will be the maximum

of the scores obtained in the respective two years.

A student who is going to repeat the first year of the B. Math (Hons) course should un-

dergo counseling by the Dean of Studies/Students-In-charge in the presence of his/her par-

ents/guardians, to assess whether the student has an aptitude for the programme.

2.6 Final Result

At the end of the third academic year the overall average of the percentage composite scores in

all the credit courses taken in the three-year programme is computed for each student. Each

of the credit courses carries a total of 100 marks. The student is awarded the B. Math (Hons.)

degree in one of the following categories according to the criteria she/he satisfies, provided

his/her conduct is satisfactory, and she/he passes all the semesters.

• B. Math (Hons.) - First Division with distinction

(i) The overall average score is at least 75%,

(ii) average score in the sixteen core courses 2 is at least 60%, and

(iii) the number of composite scores less than 45% is at most one.

2The 16 core courses are Algebra I, II, III, IV; Analysis I, II, III, IV; Probability I, II; Optimization, Com-

plex Analysis, Graph Theory, Topology, Introduction to Differential Geometry and Introduction to Differential

Equations. See Section 3.2 for detailed syllabi.

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• B. Math (Hons.)- First Division

(i) Not in the First Division with distinction

(ii) The overall average score is at least 60%,

(iii) average score in the sixteen core courses is at least 60%, and

(iv) the number of composite scores less than 45% is at most four.

• B. Math (Hons.)- Second Division

(i) Not in the First Division with distinction or First Division,

(ii) the overall average score is at least 45%,

(iii) average score in the sixteen core courses is at least 45%, and

(iv) the number of composite scores less than 45% is at most six.

If a student has satisfactory conduct, passes all the courses but does not fulfill the requirements

for the award of the degree with Honours, then she/he is awarded the B. Math degree without

Honours. A student fails if his/her composite score in any credit or non- credit course is less

than 35%.

2.7 Award of Certificates

A student passing the B. Math degree examination is given a certificate which includes (i) the

list of all the credit courses taken in the three-year programme along with their respective

composite scores and (ii) the category (Hons. First Division with Distinction or Hons. First

Division or Hons. Second Division or Pass) of his/her final result.

The Certificate is awarded in the Annual Convocation of the Institute following the last semes-

tral examinations.

2.8 Fees, Stipend, and Contigency Grant

Other than refundable Library and Hostel deposit and the recurring mess fees there are no fees

charged by the institute.

A monthly Stipend of Rs 3000, is awarded at the time of admission to each student. This is

valid initially for the first semester only. A repeating student will not get any stipend or contin-

gency grant or prizes during the repeat year. However, if she/he is from such an economically

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underprivileged background that this step will force him/her to discontinue, then she/he can

appeal to the Dean of Studies or the Students In-charge, for financial support. The amount of

stipend to be awarded in each subsequent semester depends on academic performance, conduct,

and attendance, as specified below, provided the requirements for continuation in the academic

programme (excluding repetition) are satisfied; see Sections 2.3 and 1.5.

1. Students having Scholarships: If a student is getting a scholarship from another gov-

ernment agency then the stipend will be discontinued. If during the B. Math (hons.)

programme the student obtains any scholarship with retrospective effect then the student

should return the stipend given by the institute. Failure to do so will be deemed as

unsatisfactory conduct and corresponding rules shall apply.

2. Performance in course work

If, in any particular semester, (i) the composite score in any course is less than 35%,

or (ii) the composite score in more than one course (two courses in the case of the first

semester of the first year) is less than 45%, or (iii) the average composite score in all

credit courses is less than 45%, no stipend is awarded in the following semester.

If all the requirements for continuation of the programme are satisfied, the average com-

posite score is at least 60% and the number of credit course scores less than 45% is at

most one in any particular semester (at most two in the first semester of the first year),

the full value of the stipend is awarded in the following semester.

If all the requirements for continuation of the programme are satisfied, the average com-

posite score is at least 45% but less than 60%, and the number of credit course scores less

than 45% is at most one in any particular semester (at most two in the first semester of

the first year), the stipend is halved in the following semester.

All composite scores are considered after the respective back-paper examinations. Stipend

is fully withdrawn as soon as the requirements for continuation in the academic programme

are not met.

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3. Attendance

If the overall attendance in all courses in any semester is less than 75%, no stipend is

awarded in the following semester.

4. Conduct

The Dean of Studies or Associate Dean or Students-In-Charge or the Class Teacher, at

any time, in consultation with the respective Teachers’ Committee, may withdraw the

stipend of a student fully for a specific period if his/her conduct in the campus is found

to be unsatisfactory.

Once withdrawn, stipends may be restored in a subsequent semester based on improved per-

formance and/or attendance, but no stipend is restored with retrospective effect.

Stipends are given after the end of each month for twelve months in each academic year. The

first stipend is given two months after admission with retrospective effect provided the student

continues in the B. Math (Hons.) programme for at least two months.

An yearly contingency grant of Rs 3000 is given to students at the time of admission. Con-

tingency grants can be used for purchasing a scientific calculator (or calculator) and other

required accessories for the practical class, text books and supplementary text books and for

getting photocopies of required academic material. All such expenditure should be approved

by the Students-In-Charge. Contingency grants can be utilised after the first two months of

admission. Every student is required to bring a scientific calculator for use in the practical

classes.

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3 B. Math (Hons.) Curriculum and Detailed Syllabi

3.1 Curriculum

Writing of Mathematics will be allocated two one hour lecture sessions per week. All other

courses are allocated four one hour lecture sessions per week. Statistics, Physics, and Computer

Science courses may be allocated an extra one hour session per week to facilitate Laboratory

work.

First Year

Semester I Semester II

1. Analysis I (Calculus of one variable) 1. Analysis II (Metric spaces and Multivariate

Calculus)

2. Probability Theory I 2. Probability Theory II

3. Algebra I (Groups) 3. Algebra II (Linear Algebra)

4. Computer Science I (Programming) 4. Physics I (Mechanics of particles and

5. Writing of Maths (non-credit half-course) Continuum systems)

Second Year

Semester I Semester II

1. Analysis III (Vector Calculus) 1. Graph Theory

2. Algebra III (Rings and Modules) 2. Algebra IV (Field Theory)

3. Statistics I 3. Statistics II

4. Physics II (Thermodynamics and Optics) 4. Topology

5. Optimization 5. Computer Science II (Numerical Methods)

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Third Year

Semester I Semester II

1. Complex Analysis 1. Analysis IV (Introduction to Function Spaces)

2. Introduction to Differential Geometry 2. Introduction to Differential Equations

3. Physics III 3. Physics IV (Modern Physics and Quantum

Mechanics)

4. Statistics III 4. Elective Subject II

5. Elective Subject I 5. Elective Subject III

3.2 Sixteen Core courses

Algebra I (Groups) : Basic set theory: Equivalence relations and partitions. Zorn’s lemma.

Axiom of choice. Principle of induction. Groups, subgroups, homomorphisms. Modular

arithmetic, quotient groups, isomorphism theorems. Groups acting on sets. Sylow’s

theorems. Permutation groups. Semidirect products, Free groups.

Algebra II (Linear Algebra) : Matrices and determinants. Linear equations. Basics of

Polynomial rings over real and complex numbers. Vector spaces over fields. Bases and

dimensions. Direct sums and quotients of vector spaces. Linear transformations and

their matrices. Eigenvalues and eigen vectors. Characteristic polynomial and minimal

polynomial. Bilinear forms, inner products, symmetric, hermitian forms. Unitary and

normal operators. Spectral theorems.

Algebra III (Rings and Modules) : Ring homomorphisms, quotient rings, adjunction of

elements. Polynomial rings. Chinese remainder theorem and applications. Factorisation

in a ring. Irreducible and prime elements, Euclidean domains, Principal Ideal Domains,

Unique Factorisation Domains. Field of fractions, Gauss’s lemma. Noetherian rings,

Hilbert basis theorem. Finitely generated modules over a PID and their representation

matrices. Structure theorem for finitely generated abelian groups. Rational form and

Jordan form of a matrix.

Algebra IV (Field Theory) : Finite Fields. Field extensions, degree of a field extension.

Ruler and compass constructions. Algebraic closure of a field. Transcendental bases. Ga-

lois theory in characteristic zero, Kummer extensions, cyclotomic extensions, impossibil-

13

ity of solving quintic equations. Time permitting: Galois theory in positive characteristic

(separability, normality), Separable degree of an extension.

Reference Texts : 1. M. Artin: Algebra.

2. S. D. Dummit and M. R. Foote: Abstract Algebra.

3. I. N. Herstein: Topics in Algebra.

4. K. Hoffman and R. Kunze: Linear Algebra.

Analysis I (Calculus of one variable) : The language of sets and functions - countable and

uncountable sets (see also Algebra 1). Real numbers - least upper bounds and greatest

lower bounds. Sequences - limit points of a sequence, convergent sequences; bounded

and monotone sequences, the limit superior and limit inferior of a sequence. Cauchy

sequences and the completeness of R. Series - convergence and divergence of series, ab-

solute and conditional convergence. Various tests for convergence of series. (Integral test

to be postponed till after Riemann integration is introduced in Analysis II.) Connection

between infinite series and decimal expansions, ternary, binary expansions of real num-

bers, calculus of a single variable - continuity; attainment of supremum and infimum of

a continuous function on a closed bounded interval, uniform continuity. Differentiability

of functions. Rolle’s theorem and mean value theorem. Higher derivatives, maxima and

minima. Taylor’s theorem - various forms of remainder, infinite Taylor expansions.

Analysis II (Metric spaces and Multivariate Calculus) : The existence of Riemann in-

tegral for sufficiently well behaved functions. Fundamental theorem of Calculus. Calculus

of several variables: Differentiability of maps from Rm to Rn and the derivative as a lin-

ear map. Higher derivatives, Chain Rule, Taylor expansions in several variables, Local

maxima and minima, Lagrange multiplier. Elements of metric space theory - sequences

and Cauchy sequences and the notion of completeness, elementary topological notions for

metric spaces i.e. open sets, closed sets, compact sets, connectedness, continuous and

uniformly continuous functions on a metric space. The Bolzano - Weirstrass theorem,

Supremum and infimum on compact sets, Rn as a metric space.

Analysis III (Vector Calculus) : Multiple integrals, Existence of the Riemann integral for

sufficiently well-behaved functions on rectangles, i.e. product of intervals. Multiple in-

tegrals expressed as iterated simple integrals. Brief treatment of multiple integrals on

14

more general domains. Change of variables and the Jacobian formula, illustrated with

plenty of examples. Inverse and implicit functions theorems (without proofs). More ad-

vanced topics in the calculus of one and several variables - curves in R2 and R3. Line

integrals, Surfaces in R3, Surface integrals, Divergence, Gradient and Curl operations,

Green’s, Strokes’ and Gauss’ (Divergence) theorems. Sequence of functions - pointwise

versus uniform convergence for a function defined on an interval of R, integration of a

limit of a sequence of functions. The Weierstrass’s theorem about uniform approximation

of a continuous function by a sequence of polynomials on a closed bounded interval.

Analysis IV (Introduction to Function Spaces) : Review of compact metric spaces.

C([a, b]) as a complete metric space, the contraction mapping principle. Banach’s contrac-

tion principle and its use in the proofs of Picard’s theorem, inverse and implicit function

theorems. The Stone-Weierstrass theorem and Arzela-Ascoli theorem for C(X). Peri-

odic functions, Elements of Fourier series - uniform convergence of Fourier series for well

behaved functions and mean square convergence for square integrable functions.

Reference Texts : 1. T. M. Apostol: Mathematical Analysis.

2. T. M. Apostol: Calculus

3. S. Dineen: Multivariate Calculus and Geometry.

4. R. R. Goldberg: Methods of Real Analysis.

5. T. Tao: Analysis I.

6. Bartle and Sherbert: Introduction to Real Analysis.

7. H. Royden: Real Analysis.

Introduction to Differential Equations : Ordinary differential equations - first order equa-

tions, Picard’s theorem (existence and uniqueness of solution to first order ordinary differ-

ential equation), Second order linear equations - second order linear differential equations

with constant coefficients, Systems of first order differential equations, Equations with

regular singular points, Introduction to power series and power series solutions, Special

ordinary differential equations arising in physics and some special functions (eg. Bessel’s

functions, Legendre polynomials, Gamma functions). Partial differential equations - el-

ements of partial differential equations and the three equations of physics i.e. Laplace,

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Wave and the Heat equations, at least in 2-dimensions. Lagrange’s method of solving first

order quasi linear equations.

Reference Texts : 1. G.F. Simmons: Differential Equations.

2. R. Haberman: Elementary applied partial differential equations.

3. R. Dennemeyer: Introduction to partial differential equations and boundary value

problems.

Complex Analysis : Holomorphic functions and the Cauchy-Riemann equations, Power se-

ries, Functions defined by power series as holomorphic functions, Complex line integrals

and Cauchy’s theorem, Cauchy’s integral formula. Representations of holomorphic func-

tions in terms of power series. Zeroes of analytic functions, Liouville’s theorem, The fun-

damental theorem of algebra, The maximum modulus principle, Schwarz’s lemma, The

argument principle, The open mapping property of holomorphic functions. The calculus

of residues and evaluation of integrals using contour integration.

Reference Texts : 1. D. Sarason: Notes on Complex Function Theory.

2. T. W. Gamelin: Complex Analysis.

3. J. B. Conway: Functions of one complex Variable.

Topology : Topological spaces, quotient topology. Separation axioms, Urysohn lemma. Con-

nectedness and compactness. Tychonff’s theorem, one point compactification.

Reference Texts : 1. J. Munkres: Topology a first course.

2. M. A. Armstrong: Basic Topology.

3. G. F. Simmons: Introduction to Topology and Modern Analysis.

4. K. Janich: Topology.

Introduction to Differential Geometry : Curves in two and three dimensions, Curvature

and torsion for space curves, Existence theorem for space curves, Serret-Frenet formula for

space curves, Inverse and implicit function theorems, Jacobian theorem, Surfaces in R3 as

two dimensional manifolds, Tangent space and derivative of maps between manifolds, First

fundamental form, Orientation of a surface, Second fundamental form and the Gauss map,

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Mean curvature and scalar curvature, Integration on surfaces, Stokes formula, Gauss-

Bonnet theorem.

Reference Texts : 1. M.P. do Carmo: Differential Geometry of Curves and Surfaces.

2. A. Pressley: Elementary Differential Geometry.

Graph Theory : Graphs, Hamilton Cycles and Euler Cycles, Planar Graphs, vector spaces

and matrices associated with Graphs, Flows in Directed Graphs, Connectivity and

Menger’s Theorem, Matching, Tutte’s 1-Factor Theorem.

Reference Texts : 1. B. Bollobas: Graph Theory (Chapters I - III).

2. P. J. Cameron and J.H. Van Lint: Graphs, codes and designs.

Optimization : Systems of linear equations: Gaussian elimination, LU, QR decompositions,

Singular values, SVD, Inner product spaces, Projections onto subspaces, Perron- Frobe-

nius, Fundamental theorem of LP, The simplex algorithm, Duality and applications, LP

and Game theory.

Reference Texts : 1. Harry Dym: Linear algebra in action, AMS Publications 2011.

2. A. R. Rao and Bhimasankaram: Linear Algebra

3. G. Strang: Applied linear algebra.

4. C. R. Rao: Linear statistical inference.

5. H. Karloff: Linear programming.

6. S-C Fang and S.Puthenpura: Linear optimization and extensions.

Probability I : Orientation, Combinatorial probability and urn models, Independence of

events, Conditional probabilities, Random variables, Distributions, Expectation, Vari-

ance and moments, probability generating functions and moment generating functions,

Standard discrete distributions (uniform, binomial, Poisson, geometric, hypergeometric),

Independence of random variables, Joint and conditional discrete distributions. Univariate

densities and distributions, standard univariate densities (normal, exponential, gamma,

beta, chi-square, cauchy). Expectation and moments of continuous random variables.

Transformations of univariate random variables. Tchebychev’s inequality and weak law

of large numbers.

17

Probability II : Joint densities and distributions. Transformation of variables (assuming Ja-

cobian formula). Distributions of sum, maxima, minima, order statistics, range etc. Mul-

tivariate normal (properties, linear combinations) and other standard multivariate distri-

butions (discrete and continuous) as examples. Standard sampling distributions like t,

χ2 and F . Conditional distributions, Conditional Expectation. Characteristic functions:

properties, illustrations, inversion formula, continuity theorem (without proof). Central

Limit Theorem for i.i.d. case with finite variance. Elements of modes of convergence of

random variables and the statement of the strong law of large numbers.

Reference Texts : 1. K. L. Chung: Elementary Probability Theory.

2. P. G. Hoel, S. C. Port and C. J. Stone : Introduction to Probability Theory.

3. R. Ash : Basic Probability Theory.

4. W. Feller : Introduction to Probability Theory and its Applications, Volume 1.

5. W. Feller : Introduction to Probability Theory and its Applications, Volume 2.

6. P. Billingsley : Probability and Measure.

7. V. K. Rohatgi: Probability theory.

3.3 Statistics courses

Statistics I : Introduction to Statistics with examples of its use; Descriptive statistics; Graph-

ical representation of data: Histogram, Stem-leaf diagram, Box-plot; Exploratory statis-

tical analysis with a statistical package; Basic distributions, properties; Model fitting

and model checking: Basics of estimation, method of moments, Basics of testing, in-

terval estimation; Distribution theory for transformations of random vectors; Sampling

distributions based on normal populations: t, χ2 and Fx distributions. Bivariate data,

covariance, correlation and least squares

Suggested References: 1. Lambert H. Koopmans: An introduction to contemporary

statistics

2. David S Moore and William I Notz: Statistics Concepts and Controversies

3. David S Moore, George P McCabe and Bruce Craig: Introduction to the Practice of

Statistics

18

4. Larry Wasserman: All of Statistics. A Concise Course in Statistical Inference

5. John A. Rice: Mathematical Statistics and Data Analysis

Statistics II : Theory and Methods of Estimation and Hypothesis testing, Sufficiency, Ex-

ponential family, Bayesian methods, Moment methods, Maximum likelihood estimation,

Criteria for estimators, UMVUE, Large sample theory: Consistency; asymptotic normal-

ity, Confidence intervals, Elements of hypothesis testing; Neyman-Pearson Theory, UMP

tests, Likelihood ratio and related tests, Large sample tests.

Suggested References : 1. George Casella and Roger L Berger: Statistical Inference

2. Peter J Bickel and Kjell A Doksum: Mathematical Statistics

3. Erich L Lehmann and George Casella: Theory of Point Estimation

4. Erich L Lehmann and Joseph P Romano: Testing Statistical Hypotheses

Statistics III : Multivariate normal distribution, Transformations and quadratic forms; Re-

view of matrix algebra involving projection matrices and matrix decompositions; Linear

models; Regression and Analysis of variance; General linear model, Matrix formulation,

Estimation in linear model, Gauss-Markov theorem, Estimation of error variance, Testing

in the linear model, Regression, Partial and multiple correlations, Analysis of variance,

Multiple comparisons; Stepwise regression, Regression diagnostics.

Suggested References: 1. Sanford Weisberg: Applied Linear Regression

2. C R Rao: Linear Statistical Inference and Its Applications

3. George A F Seber and Alan J Lee: Linear Regression Analysis

3.4 Computer Science Courses

Computer Science I (Programming) Recommended Language: C Basic abilities of

writing, executing, and debugging programs. Basics: Conditional statements, loops,

block structure, functions and parameter passing, single and multi-dimensional arrays,

structures, pointers. Data Structures: stacks, queues, linked lists, binary trees. Simple

algorithmic problems: Some simple illustrative examples, parsing of arithmetic expres-

sions, matrix operations, searching and sorting algorithms.

19

Computer Science II (Numerical Methods) Introduction to Matlab (or appropriate

package) and Numerical Computing: Number representations, finite precision arithmetic,

errors in computing. Convergence, iteration, Taylor series. Solution of a Single Non-linear

Equation: Bisection method. Fixed point methods. Newton’s method. Convergence to

a root, rates of convergence. Review of Applied Linear algebra: Vectors and matrices.

Basic operations, linear combinations, basis, range, rank, vector norms, matrix norms.

Special matrices. Solving Systems of equations (Direct Methods): Linear systems. So-

lution of triangular systems. Gaussian elimination with pivoting. LU decomposition,

multiple right-hand sides. Nonlinear systems. Newton’s method. Least Squares Fitting

of Data: Fitting a line to data. Generalized least squares. QR decomposition. Interpo-

lation: Polynomial interpolation by Lagrange polynomials. Alternate bases: Monomials,

Newton, divided differences. Piecewise polynomial interpolation. Cubic Hermite poly-

nomials and splines. Numerical Quadrature: Newton - Cotes Methods: Trapezoid and

Simpson quadrature. Gaussian quadrature. Adaptive quadrature. Ordinary Differential

Equations: Euler’s Method. Accuracy and Stability. Trapezoid method. Runge - Kutta

method. Boundary value problems and finite differences.

Reference Texts: 1. B. Kernighan and D. Ritchie: The C Programming Language.

2. J. Nino and F. A. Hosch: An Introduction to Programming and Object Oriented

Design using JAVA.

3. G. Recketenwald: Numerical Methods with Matlab.

4. Shilling and Harries: Applied Numerical methods for engineers using Matlab and C.

5. S. D. Conte and C. De Boor: Elementary Numerical Analysis: An Algorithmic

Approach.

6. S. K. Bandopadhyay and K. N. Dey: Data Structures using C.

7. J. Ullman and W. Jennifer: A first course in database systems.

3.5 Physics Courses

Physics I (Mechanics of Particles and Continuum Systems): Newton’s laws of mo-

tion; Concept of inertial frames of reference. Conservation laws (energy, linear momentum,

angular momentum) for a single particle and a system of particles; Motion of system with

20

variable mass; Frictional forces; Center of mass and its motion; Simple collision problems;

Torque; Moment of Inertia (parallel and perpendicular axis theorem) and Kinetic energy

of a rotating rigid body; Central forces; Newtons laws of gravitation; Keplers laws; Ele-

ments of Variational calculus and Lagrangian formulation; Introduction to mechanics of

continuum systems; Elastic deformation and stress in a solid; Hookes law; 9 interrelations

of elastic constants for an isotropic solid; elastic waves; basic elements of fluid dynam-

ics; equation of continuity; Eulers equation for ideal fluid; Streamline flow; Bernoullis

equation +5 experiments (10 hours)

Suggested Text Books: 1. Mechanics - Keith Symon :

2. Classical Mechanics- R. Douglas Gregory

3. Classical Mechanics - J. R. Taylor

Physics II ( Thermal physics and Optics): Kinetic theory of Gases; Ideal Gas equation;

Maxwells Laws for distribution of molecular speeds. Introduction to Statistical mechanics;

Specification of state of many particle system; Reversibility and Irreversibility ; Behavior

of density of states; Heat and Work; Macrostates and Microstates; Quasi static processes;

State function; Exact and Inexact differentials; First Law of Thermodynamics and its

applications; Isothermal and Adiabatic changes; reversible, irreversible, cyclic processes;

Second law of thermodynamics; Carnots cycle. Absolute scale of temp; Entropy; Joule

Thomson effect; Phase Transitions; Maxwells relations; Connection of classical thermo-

dynamics with statistical mechanics; statistical interpretation of entropy.; Third law of

Thermodynamics

Optics:

Light as a scalar wave; superposition of waves and interference; Youngs double slit exper-

iment; Newtons rings; Thin films; Diffraction; Polarization of light; transverse nature of

light waves

Texts: 1. F. Reif: Statistical and Thermal physics

2. Kittel and Kroemer: Thermal Physics

3. Zeemansky and Dittman: Heat and Thermodynamics

4. Jenkins and White: Optics

21

Physics III (Electromagnetism and Electrodynamics) Vectors, Vector algebra, Vector

Calculus (Physical meaning of gradient, divergence and curl); Gauss’s divergence theo-

rem; Theorems of gradients and curls. Electrostatics, Coulomb’s law for discrete and con-

tinuous charge distribution; Gauss’s theorem and its applications; Potential and field due

to simple arrangements of electric charges; work and energy in electrostatics; Dilectrics,

Polarization; Electric displacement; Capacitors (Paralleleplates); Electrical images. Mag-

netostatics: Magnetic field intensity (H), Magnetic induction (B), Biot-Savart’s law; Am-

pere’s law; comparison of electrostatics and magnetostatics. Electro- dynamics: Ohm’s

law, Electromotive force, Faraday’s law of electro- magnetic induction; Lorentz force,

Maxwell’s equations. Electromagnetic theory of light and wave optics. Electronics: Semi-

conductors; pn junctions; transistors; zenor diode, IV characteristics. +5 experiments (10

hours)

Reference Texts: 1. D. J. Giffths: Introduction to Electrodynamics.

2. J. R. Reitz, F. J. Milford and W. Charisty: Foundations of Electromagnetic theory.

10

3. D. Halliday and R. Resnick: Physics II.

Physics IV (Modern Physics and Quantum Mechanics): Special theory of Relativity:

Michelson-Morley Experiment, Einstein’s Postulates, Lorentz Transformations, length

contraction, time dilation, velocity transformations, equivalence of mass and energy. Black

body Radiation, Planck’s Law, Dual nature of Electromagnetic Radiation, Photoelec-

tric Effect, Compton effect, Matter waves, Wave-particle duality, Davisson-Germer ex-

periement, Bohr’s theory of hydrogen spectra, concept of quantum numbers, Frank-Hertz

experiment, Radioactivity, X-ray spectra (Mosley’s law), Basic assumptions of Quantum

Mechanics, Wave packets, Uncertainty principle, Schrodinger’s equation and its solution

for harmonic oscillator, spread of Gaussian wave packets with time.

A list of possible physics experiments:

1. Determination of the coefficient of viscosity of water by Poiseuille’s method (The diameter

of the capillary tube to be measured by a travelling vernier microscope).

2. Determination of the surface tension of water by capillary rise method.

22

3. Determination of the temp. coefficient of the material of a coil using a metric bridge.

4. To draw the frequency versus resonant length curve using a sonometer and hence to find

out the frequency of the given tuning fork.

5. Study of waves generated in a vibrating string and vibrating membrane.

6. Determination of wave length by Interference and Diffraction.

7. One experiment on polarized light.

8. Experiments on rotation of place of polarization - chirality of media.

9. Elasticity: Study of stress-strain relation and verification of Hooke’s law of elasticity,

Measurement of Young’s modulus.

10. Faraday Experiment: Pattern in fluid and granular materials under parametric oscillation.

11. Determination of dispersion rotation of Faraday waves in liquid (water/glucerol) and to

compute the surface tension of the liquid.

12. Determination of the moment of a magnet and the horizontal components of Earth’s

magnetic field using a deflection and an oscillation magnetometer.

13. Familiarization with components, devices and laboratory instruments used in electronic

systems.

14. To study the characteristics of a simple resistor-capacitor circuit.

15. Transistor Amplifier: To study a common emitter bipolar junction transistor amplifier.

16. Diodes and Silicon controlled rectifiers: To study the operational characteristics of diodes

and silicon controlled rectifier.

17. Logic circuits: Combinational logic and binary addition.

3.6 Non Credit Courses

Writing of Mathematics (non-credit half-course) The aim of this (non-credit) course is to

improve the writing skills of students while inculcating an awareness of mathematical

history and culture. The instructor may choose a book, like the ones listed below, and

organize class discussions. Students will then be assigned five formal writing assignments

(of 8 to 10 pages each) related to these discussions. These will be corrected, graded and

returned.

23

Reference Texts: 1. J. Stillwell: Mathematics and Its History, Springer UTM.

2. W. Dunham, Euler: The Master of Us All, Mathematical Association of America.

3. W. Dunham: Journey Through Genius, Penguin Books.

4. M. Aigner and M. Ziegler: Proofs From The Book, Springer.

5. A. Weil: Number Theory, An Approach Through History from Hammurapi to Leg-

endre.

3.7 Elective Courses

Introduction to Representation Theory: Introduction to multilinear algebra: Review of

linear algebra, multilinear forms, tensor products, wedge product, Grassmann ring, sym-

metric product. Representation of finite groups: Complete reducibility, Schur’s lemma,

characters, projection formulae. Induced representation, Frobenius reciprocity. Repre-

sentations of permutation groups.

Reference Texts: 1. W. Fulton and J. Harris: Representation Theory, Part I.

2. J-P Serre: Linear representations of finite groups.

Introduction to algebraic geometry: Prime ideals and primary decompositions, Ideals in

polynomial rings, Hilbert basis theorem, Noether normalisation theorem, Hilbert’s Null-

stellensatz, Projective varieties, Algebraic curves, Bezout’s theorem, Elementary dimen-

sion theory.

Reference Texts: 1. M. Atiyah and I.G. MacDonald: Commutative Algebra.

2. J. Harris: Algebraic Geometry.

3. I. Shafarervich: Basic Algebraic Geometry.

4. W. Fulton: Algebraic curves.

5. M. Ried: Undergraduate Commutative Algebra.

Introduction to Algebraic Number Theory: Number fields and number rings, prime de-

composition in number rings, Dedekind domains, definition of the ideal class group, Galois

theory applied to prime decomposition and Hilbert’s ramification theory, Gauss’s reci-

procity law, Cyclotomic fields and their ring of integers as an example, the finiteness of

the ideal class group, Dirichlet’s Unit theorem.

24

Reference Texts: 1. D. Marcus: Number fields.

2. G. J. Janusz: Algebraic Number Theory.

Differential Geometry II: Manifolds and Lie groups, Frobenius theorem, Tensors and Dif-

ferential forms, Stokes theorem, Riemannian metrics, Levi-Civita connection, Curvature

tensor and fundamental forms.

Reference Texts: 1. S. Kumaresan: A course in Differential Geometry and Lie Groups.

2. T. Aubin: A course in Differential Geometry.

Introduction to Differential Topology: Manifolds. Inverse function theorem and immer-

sions, submersions, transversality, homotopy and stability, Sard’s theorem and Morse

functions, Embedding manifolds in Euclidean space, manifolds with boundary, intersec-

tion theory mod 2, winding numbers and Jordan- Brouwer separation theorem, Borsuk-

Ulam fixed point theorem.

Reference Texts: 1. V. Guillemin and Pollack: Differential Topology (Chapters I, II and

Appendix 1, 2).

2. J. Milnor: Topology from a differential viewpoint.

Topics in Optimization: Constrained optimization problems, Equality constraints, La-

grange multipliers, Inequality constraints, Kuhn-Tucker theorem, Convexity. If time per-

mits :-

Calculus of variations, Optimal control, Game theory.

References Texts: 1. R. K. Sundaram, A first course in Optimization theory.

2. S. Tijs, Introduction to game theory.

3. Fleming and Rishel, Deterministic and stochastic optimal control.

Combinatorics: Review of finite fields, Mutually orthogonal Latin squares and finite pro-

jective planes, Desargues’s theorem, t-designs and their one-point extensions, Review of

group actions - transitive and multiply transitive actions, Mathieu groups, Witt designs,

Fisher’s inequality, Symmetric designs.

25

Reference Texts: 1. D. R. Hughes and F. Piper: Projective planes, Graduate texts in

Mathematics 6.

2. P.J.Cameron and J.H.van Lint, Graphs, codes and designs.

Topics in Applied Stochastic Processes: Discrete parameter martingales (without condi-

tional expectation w.r.t. σ algebras!), Branching processes, Markov models for epidemics,

Queueing models.

Notes: (i) Measure theory to be avoided. Of course, use of DCT, MCT, Fubini, etc.

overtly or covertly permitted. (ii) Relevant materials concerning Markov chains, including

continuous time MC’s, may be reviewed. If this course runs concurrently with Prob. III

(where Markov chains are taught), some concepts/facts needed may be stated with proofs

deferred to Prob. III. (iii) Genetic models may also be included, but then at least two

topics from the above may have to deleted, as the background material from genetics may

be formidable.

References: 1. A. Goswami and B. V. Rao: A Course in Applied Stochastic Processes.

Hindustan Book Agency

2. S. Karlin and H. M. Taylor: A First and Second Course in Stochastic Processes.

Academic Press, 1975 and 1981.

3. S. M. Ross: Introduction to Probability Models. 8th edition. Academic

Press/Elsevier, Indian reprint, 2005. (Paperback)

4. S. M. Ross: Stochastic Processes. 2nd edition. Wiley Student Edition, 2004. (Pa-

perback)

Introduction to Dynamical systems: Linear maps and linear differential equation: attrac-

tors, foci, hyperbolic points; Lyapunov stability criterion, Smooth dynamics on the plane:

Critical points, Poincare index, Poincare- Bendixon theorem, Dynamics on the circle: Ro-

tations: recurrence, equidistribution, Invertible transformations: rotation number, Denjoy

construction, Conservative systems: Poincare recurrence. Newtonian mechanics.

Reference Texts: 1. B. Hasselblatt and A. Katok: A first course in dynamics.

2. M. Brin and G. Stuck: Introduction to dynamical systems.

3. V. I. Arnold: Geometrical methods in the theory of Ordinary Differential Equations.

26

Introduction to Stochastic Processes: Discrete Markov chains with countable state space.

Classification of states - recurrence, transience, periodicity. Stationary distributions,

reversible chains. Several illustrations including the Gambler’s Ruin problem, queuing

chains, birth and death chains etc. Poisson process, continuous time markov chain with

countable state space, continuous time birth and death chains.

Reference Texts: 1. P. G. Hoel, S. C. Port and C. J. Stone: Introduction to Stochastic

Processes.

2. S. M. Ross: Stochastic Processes.

3. J. G. Kemeny, J. L. Snell and A. W. Knapp: Finite Markov Chains.

4. D. L. Isaacsen and R. W. Madsen: Markov Chains, Theory and Applications.

Stochastic Models in Insurance: 1. Review of Markov chains, Poisson processes

2. Renewal processes (Basics, Renewal equations; Blackwell’s renewal theorem may be

stated without proof)

3. Claim size distributions

4. Cramer-Lundberg and Renewal risk models.

5. Ruin problems

6. Markov chain methods in life-insurance

7. Some discussion on statistical methods

Reference Texts: 1. T. Mikosch: Non-life insurance mathematics. Springer(India), 2004.

(Paperback)

2. H. U. Gerber: Life-insurance mathematics. Springer (India), 2010. (Paperback)

3. W. Feller: An introduction to probability theory, vol.II. Wiley-Eastern. (Paperback)

4. T.Rolski, Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt, Jozef Teugels,

Stochastic Processes for insurance and finance. Wiley. 1999

5. P.Boland: Statistical and probabilistic methods in actuarial science. Chapmen and

Hall, 2007.

6. R.Hogg and S.Klugman: Loss distributions. Wiley. 1984.

27

Elements of Statistical Computing: Examples and use of computational techniques in

data analysis; Simulations; Monte-Carlo sampling; E-M algorithm; Markov chain Monte-

Carlo methods, Gibbs sampling, Hastings algorithm, reversible jump MCMC; Resampling

methods: Jackknife, Bootstrap, Cross-validation.

Reference Texts: 1. Christian P Robert and George Casella: Monte Carlo Statistical

Methods

2. Brian D Ripley: Stochastic Simulation

3. Geoffrey J McLachlan and T Krishnan: The EM Algorithm and Extensions

4. Sheldon Ross: Simulation

5. B Efron: The Jackknife, the Bootstrap, and Other Resampling Plans.

Statistics IV : Analysis of Discrete data: Nonparametric methods: Decision theory, Good-

ness of fit tests, Multiway contingency tables, Odds ratios, Logit model, Wilcoxon test,

Wilcoxon signed rank test, Kolmogorov test. Elements of decision theory : Bayes and

minimax procedures.

Reference Texts: 1. G. K. Bhattacharya and R. A. Johnson: Statistics : Principles and

Methods.

2. P. J. Bickel and K. A. Doksum: Mathematical Statistics.

3. E. J. Dudewicz and S. N. Mishra: Modern Mathematical Statistics.

4. V. K. Rohatgi: Introduction to Probability Theory and Mathematical Statistics.

Statistics V: Sample Surveys (1/2 Semester): Scientific basis of sample surveys. Com-

plete enumeration vs. sample surveys. Principal steps of a sample survey; illustrations,

N.SS., Methods of drawing a random sample. SRSWR and SRSWOR: Estimation, sam-

ple size determination. Stratified sampling; estimation, allocation, illustrations. System-

atic sampling, linear and circular, variance estimation. Some basics of PPS sampling,

Two-stage sampling and Cluster sampling. Nonsampling errors. Ration and Regression

methods.

Reference Texts: 1. W. G. Cochran: Sampling Techniques.

2. M. N. Murthy: Sampling Theory and Methods.

28

3. P. Mukhopadhyay: Theory and Methods of Survey Sampling.

Design of Experiments (1/2 semester): The need for experimental designs and ex-

amples, basic principles, blocks and plots, uniformity trials, use of completely randomized

designs. Designs eliminating heterogeneity in one direction: General block designs and

their analysis under fixed effects model, tests for treatment contrasts, pairwise compar-

ison tests; concepts of connectedness and orthogonality of classifications with examples;

randomized block designs and their use. Some basics of full factorial designs. Practicals

using statistical packages.

Reference Texts: 1. A. Dean and D. Voss: Design and Analysis of Experiments.

2. D. C. Montgomery: Design and Analysis of Experiments.

3. W. G. Cochran and G. M. Cox: Experimental Designs.

4. O. Kempthorne: The Design and Analysis of Experiments.

5. A. Dey: Theory of Block Designs.

Mathematics of Computation: Models of computation (including automata, PDA). Com-

putable and non-computable functions, space and time complexity, tractable and in-

tractable functions. Reducibility, Cook’s Theorem, Some standard NP complete Prob-

lems: Undecidability.

Computer Science III (Data Structures): Fundamental algorithms and data structures

for implementation. Techniques for solving problems by programming. Linked lists,

stacks, queues, directed graphs. Trees: representations, traversals. Searching (hashing,

binary search trees, multiway trees). Garbage collection, memory management. Internal

and external sorting.

Computer Science IV (Design and Analysis of Algorithms): Efficient algorithms for

manipulating graphs and strings. Fast Fourier Transform. Models of computation, in-

cluding Turing machines. Time and Space complexity. NP-complete problems and unde-

cidable problems.

Reference Texts: 1. A. Aho, J. Hopcroft and J. Ullmann: Introduction to Algorithms and

Data Structures.

29

2. T. A. Standish: Data Structure Techniques.

3. S. S. Skiena: The algorithm Design Manual.

4. M. Sipser: Introduction to the Theory of Computation.

5. J.E. Hopcroft and J. D. Ullmann: Introduction to Automata Theory, Languages and

Computation.

6. Y. I. Manin : A Course in Mathematical Logic.

Mathematical Morphology and Applications Introduction to mathematical morphology:

Minkowski addition and subtraction, Structuring element and its decompositions. Fun-

damental morphological operators: Erosion, Dilation, Opening, Closing. Binary Vs

Greyscale morphological operations. Morphological reconstructions: Hit-or-Miss trans-

formation, Skeletonization, Coding of binary image via skeletonization, Morphological

shape decomposition, Morphological thinning, thickening, pruning. Granulometry, classi-

fication, texture analysis: Binary and greyscale granulometries, pattern spectra analysis.

Morphological Filtering and Segmentation: Multiscale morphological transformations,

Top-Hat and Bottom-Hat transformations, Alternative Sequential filtering, Segmentation.

Geodesic transformations and metrics: Geodesic morphology, Graph-based morphology,

City-Block metric, Chess board metric, Euclidean metric, Geodesic distance, Dilation dis-

tance, Hausdorff dilation and erosion distances. Efficient implementation of morphological

operators. Some applications of mathematical morphology.

Reference Texts: 1. J. Serra, 1982, Image Analysis and Mathematical Morphology, Aca-

demic Press London, p. 610.

2. J. Serra, 1988, Image Analysis and Mathematical Morphology: Theoretical Ad-

vances, Academic Press, p. 411

3. L. Najman and H. Talbot (Eds.), 2010, Mathematical Morphology, Wiley, p. 507.

4. P. Soille, 2003, Morphological Image Analysis, Principles and Applications, 2nd edi-

tion, Berlin: Springer Verlag.

5. N. A. C. Cressie, 1991, Statistics for Spatial Data, John Wiley.

Economics 1: Introduction to Economics: Micro and Macro Economics Micro

Economics:Welfare Economics: Supply and Demand, Elasticity; Consumption and Con-

sumer behaviour;Production and Theory of costs. Market Organisation: Competition,

30

Monopoly.

Macro Economics:National income accounting, demand and supply. Simple Keynesian

model and extensions. Consumption and Investment. Inflation and Unemployment.

Fiscal policy Money, banking and finance.

Reference Texts 1. Intermediate Microeconomics by Hal Varian.

2. Microeconomic Theory by Richard Layard and A.A. Walters

3. Microeconomics in Context by N. Goodwin, J. Harris, J. Nelson, B. Roach and M.

Torras

4. Microeconomics: behavior, institutions, and evolution by Bowles S.

5. Macroeconomics by N. G. Mankiw.

6. Macroeconomics by R Dornbusch and S Fisher.

7. Macroeconomics in context by N Goodwin, J Harris, J Nelson, B Roach and M

Torras,

Economics 2: Themes in Development Theory and Policy Topics from:

Theories of growth: historical pattern, classical models, structural models, neo-classical

and contemporary approaches. Concept of development: beyond growth (basic needs,

capabilities, freedom). Agricultural transformation. Strategies of industrial development.

Unemployment, underemployment and poverty. Population and demographic transition.

Migration and urbanisation.Trade theory and development experience. Environment and

sustainable development

Reference Texts 1. Development Economics by Michael P Todaro.

2. Leading Issues in Economic Development by G. M Meier and J E Rauch.

3. International Economics: Theory and Policy by Paul Krugman and M Obstfeld.

4. Halting Degradation of Natural Resources by J. M. Baland and J. P. Platteau.

5. Chapters 1-3 of Book I of Adam Smith s The Wealth of Nations.

6. Development economics. Princeton University Press by Ray D.

7. Increasing returns and economic progress. The Economic Journal 38, 527 542, Young

A.A. (1928).

31

8. Industrialization and the Big Push. The Journal of Political Economy 97, 1003 1026

by Murphy K.M., A. Shleifer, and R.W. Vishny.

9. Why Poverty Persists in India: A Framework for Understanding the Indian Economy.

OUP Catalogue by Eswaran M., and A. Kotwal.

Economics 3: Poverty and Inequality: theory and empirics Topics from: Concept and

measurement of inequality in incomes. Concentration of wealth. Intra-household inequal-

ity Poverty, relatively speaking. Definition and measurement of poverty. Poverty and

undernutrition. Trends in poverty and inequality in India

Reference Texts 1. Measurement of Inequality and Poverty, Oxford University Press,

1997, by S Subramanian (ed).

2. Measuring Inequality by Frank Cowell.

3. On Economic Inequality by Amartya Sen.

4. Why Poverty Persists in India: A Framework for Understanding the Indian Economy.

OUP Catalogue by Eswaran M., and A. Kotwal.

5. Growing public: Volume 1, the story: Social spending and economic growth since

the eighteenth century by Lindert P.H.

6. The Haves and the Have-Nots: A brief and idiosyncratic history of global inequality

by Milanovic B.

4 Miscellaneous

4.1 Library Rules

Any student is allowed to use the reading room facilities in the library and allowed access to

the stacks. B. Math (Hons.) students have to pay a security deposit of Rs.2000/- in order to

avail the borrowing facility. A student can borrow at most three books at a time.

Any book from the Text Book Library (TBL) collection may be issued out to a student only

for overnight or week-end provided at least one copy of that book is left in the TBL. Only one

book is issued at a time to a student. Fine is charged if any book is not returned by the due

date stamped on the issue-slip. The library rules, and other details are posted in the library.

32

4.2 Hostel Facility

The Institute has hostel for students in the Bangalore campus. However, it may not be possible

to accommodate all degree/diploma students in the hostels. Limited medical facilities are

available free of cost at Bangalore campus.

4.3 Expenses for the Field Training Programmes

All expenses for the necessary field training programmes are borne by the Institute, as per the

Institute rules.

4.4 Change of Rules

The Institute reserves the right to make changes in the above rules, course structure and the

syllabi as and when needed.

33